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Automata TheoryCS411-2015F-04
Non-Determinisitic Finite Automata
David Galles
Department of Computer Science
University of San Francisco
04-0: Non-Determinism
A Deterministic Finite Automata’s transitionfunction has exactly one transition for eachstate/symbol pair
A Non-Deterministic Finite Automata can have 0, 1or more transitions for a single state/symbol pair
Example: L = {w ∈ {a, b} : w starts with a}
Regular expression?
04-1: NFA Example
Example: L = {w ∈ {a, b} : w starts with a}
a(a+b)*
a0 1
a,b
04-2: NFA Example
Example: L = {w ∈ {a, b} : w starts with a}
a(a+b)*
a0 1
a,b
What happens if a ’b’ is seen in state q0?
The machine “crashes”, and does not acceptthe string
04-3: NFA Example
Example: L = {w ∈ {a, b} : w contains the
substring aa}
Regular Expression?
04-4: NFA Example
Example: L = {w ∈ {a, b} : w contains the
substring aa}
(a+b)*aa(a+b)*
a0 2
a,b
1a
a,b
What happens if a a is seen in state q0?
04-5: NFA Example
Example: L = {w ∈ {a, b} : w contains the
substring aa}
(a+b)*aa(a+b)*
a0 2
a,b
1a
a,b
What happens if a a is seen in state q0?
Stay in state q0, or go on to state q1Multiple Computational Paths (board example)
04-6: NFA Example
Example: L = {w ∈ {a, b} : w contains the
substring aa}
a0 2
a,b
1a
a,b
(q0, abaa) (q0, baa) (q0,aa) (q0, a) (q0, ε) reject
(q1, baa) (q1, a)
(q1, ε)
(q2, ε)
crash
reject
reject
accept
Does this machine accept abaa?
04-7: NFA Acceptance
If there is any computational path that accepts astring, then the machine accepts the string
Two ways to think about NFAs:
Magic “Oracle”, which always picks the correctpath to take
Try all possible paths
04-8: NFA Example
Example: L = {w ∈ {a, b} : w contains the
substring aa}
a0 2
a,b
1a
a,b
If a string contains aa, will there be acomputational path that accepts it?
If a string does not contain aa, will there be acomputational path that accepts it?
04-9: NFA Definition
Difference between a DFA and an NFA
DFA has exactly only transition for eachstate/symbol pairδ : (K × Σ) 7→ K
NFA has 0, 1 or more transitions for eachstate/symbol pair
04-10: NFA Definition
Difference between a DFA and an NFA
DFA has exactly only transition for eachstate/symbol pair
Transition function: δ : (K × Σ) 7→ K
NFA has 0, 1 or more transitions for eachstate/symbol pair
Transition relation: ∆ ⊆ ((K × Σ)×K)
04-11: NFA Definition
A NFA is a 5-tuple M = (K,Σ,∆, s, F )
K Set of states
Σ Alphabet
∆ : (K × Σ)×K is a Transition relation
s ∈ K Initial state
F ⊆ K Final states
04-12: Fun with NFA
Create an NFA for:
All strings over {a, b} that start with a and end withb
04-13: Fun with NFA
Create an NFA for:
All strings over {a, b} that start with a and end withb
a0
b1 3
a,b
(example compuational paths for ababb, abba, bbab)
04-14: Fun with NFA
Create an NFA for:
All strings over {0, 1} that contian 0110 or 1001
04-15: Fun with NFA
Create an NFA for:
All strings over {0, 1} that contian 0110 or 1001
0
1
3
0,11 1 1
1
01 1 1
0
0,1
1 1
0 0
04-16: ǫ-Transitions
ǫ transition consumes no input
NFA (with ǫ transitions) for (ab)*(aab)*
1 1
a1 1 1
a
a
b
b
ε
04-17: ǫ-Transitions
Create an NFA (with ǫ-transitions) for:
All strings over {a, b, c} that are missing at leastone letter. For example, aabba, cbbc, ccacc ∈ L,while abbc 6∈ L
04-18: ǫ-Transitions
Create an NFA (with ǫ-transitions) for:
All strings over {a, b, c} that are missing at leastone letter. For example, aabba, cbbc, ccacc ∈ L,while abbc 6∈ L
2
0
b,c
1
a,b
3
a,c
ε ε ε
abcb, bbab, abbab, abc
04-19: Yet More Formalism
ǫ-closure
ǫ-closure(q) = set of all states that can bereached following zero or more ǫ-transitionsfrom state q.
04-20: ǫ-closure
ǫ-closure examples
0
6
1 2
3 4 5
ε
ε
εε
εa cb
a
a
b
(quick review: What is L[M]?)
04-21: ǫ-closure
ǫ-closure examples
0
6
1 2
3 4 5
ε
ε
εε
εa cb
a
a
b
L[M] = {a, aa, ba, ca, aba}
04-22: ǫ-closure
ǫ-closure examples
0
6
1 2
3 4 5
ε
ε
εε
εa cb
a
a
b
ǫ-closure(q0) =
ǫ-closure(q3) =
ǫ-closure(q2) =
ǫ-closure(q5) =
04-23: ǫ-closure
ǫ-closure examples
0
6
1 2
3 4 5
ε
ε
εε
εa cb
a
a
b
ǫ-closure(q0) = {q0, q1, q4, q5}
ǫ-closure(q3) = {q3, q4, q5}
ǫ-closure(q2) = {q2, q6}
ǫ-closure(q5) = {q5}
04-24: ǫ-closure – Sets
ǫ-closure
ǫ-closure(q) = set of all states that can bereached following zero or more ǫ-transitionsfrom state q.
Can extend ǫ-closure to a set of states
ǫ-closure(S) =⋃{A : q ∈ S ∧ ǫ-closure(q) = A}
04-25: NFA Definition (revised)
A NFA is a 5-tuple M = (K,Σ,∆, s, F )
K Set of states
Σ Alphabet
∆ : (K × (Σ ∪ {ǫ}))×K is a Transition relation
s ∈ K Initial state
F ⊆ K Final states
04-26: NFA ⊢M
Binary relation ⊢M : What machine M yields in onestep
⊢M⊆ (K × Σ∗)× (K × Σ∗)
⊢M= {((q1, aw), (q2, w)) : q1, q2 ∈ KM , w ∈Σ∗
M, a ∈ ΣM ∪ {ǫ}, ((q1, a), q2) ∈ ∆M}
Binary relation ⊢∗
M: Transitive, reflexive closure of
⊢M
04-27: NFA Languages
L[M ] – Language defined by NFA M
L[M ] = {w : (sM , w) ⊢∗
M(f, ǫ) for some
f ∈ FM∗}
LNFA = {L : ∃NFA M,L[M ] = L}
04-28: NFA Examples
Create an NFA for the language
Give an NFA for the language L = All stringsover {0,1} that contain two pairs of adjacent 0’sseparated by an even number of symbols. So,0100110011, 01100101100101, and 01001000are in the language, but 0100100, 1011001,and 0111011 are not in the language.
04-29: NFA Examples
Create an NFA for the language
Give an NFA for the language L = All stringsover {0,1} that contain two pairs of adjacent 0’sseparated by an even number of symbols. So,0100110011, 01100101100101, and 01001000are in the language, but 0100100, 1011001,and 0111011 are not in the language.
0,1
0 0 0 0
0,1
0,1 0,1
04-30: NFA Examples
Create an NFA for the language
L = All strings over {a,b} that have an a as oneof the last 3 charaters in the string. So, a, baab,bbbab, aabbaaabb ∈ L, but bb, baabbb,bbabbbbb 6∈ L
04-31: NFA Examples
Create an NFA for the language
L = All strings over {a,b} that have an a as oneof the last 3 charaters in the string. So, a, baab,bbbab, aabbaaabb ∈ L, but bb, baabbb,bbabbbbb 6∈ L
a,b
a b b
